granted that the solution \(\mathbf {Z}^*\)  to (REF ) has a rank less than the new latent dimension \(b\)  [1]}. However, while (REF ) is convex, (REF ) is not. One can also show that if the solution \(\mathbf {Z}^*\)  to (REF ) has a rank less than the new latent dimension \(b\) , the problem (REF ) has no spurious local minima [2]}. We can also show that any stationary point \(\hat{\mathbf {Z}} = \hat{\mathbf {U}}\hat{\mathbf {V}}^\top \)  of (REF ), achieved e.g., with gradient descent, is a global optimum for (REF ) if it satisfies the following qualification condition
\(\Vert \sum _{i=1}^n \nabla _{\mathbf {Z}}\mathcal {L}(f_i(\hat{\mathbf {Z}}), \mathbf {Y}_i)\Vert _2 \le \beta ,\) 
